Operations with Integers

Materials: counters in two colors (red and black are the colors discussed in this lesson), a large classroom number line, individual number lines, chart paper

Preparation: Make up packages of equal numbers of red and black counters (or red and black squares of paper) for groups of students. There should be at least 20 of each color in a package.

Say:Yesterday, you figured out some ways to add and subtract using the number line. Today, we'll use the number line to multiply and divide.

Ask: Can someone show me, on the number line, how to multiply 3 by 5?
Students should, by now, have internalized the commutativity of multiplication, so encourage students to show both 3 x 5 and 5 x 3. Work on a description of the action that includes both of the factors, the action, and the product: Use the first factor as a unit. If the first factor is 3, the distance from 0 to 3 on the number line is our unit. Now use the second factor to tell how many of these units we have. If the second factor is 5, move 5 units (that's 5 groups of 3) to the right. You're now on the product.

Ask:Can someone show me, on the number line, how to multiply 3 by 5?
If students use the description of multiplication shown above, they'll see that if the unit is 3, the action must move to the left as they take repeated groups of negative 3.

Ask:Can someone show me, on the number line, how to multiply 3 by 5?
While it may not be as easy to visualize negative 5 of groups of 3, students can use what they know about commutativity to think of 5 as the unit and 3 as the number of groups.

Now get out the counters and show students that these are models of negative and positive numbers. The red counters are each 1. The black counters are each +1. Discuss the value of a red/black pair (0).

Say:Work in your small groups to come up with a way to show the three multiplication examples we've already done on the number line (3 x 5, 3 x 5, 3 x 5).
When students have had time to work, discuss their results. To multiply 3 by 5, they show five groups of black counters. To multiply 3 by 5, they show five groups of three red counters. To multiply 3 by 5, they show three groups of five red counters.

Say:That was all very straightforward, but how would you show how to multiply 3 by 5?
Help students to see that if they think about the repeated-addition meaning of multiplication, they can work this out. If 3 x 5 is the same as 5 + 5 + 5, then 3 x 5 is the same as 5 + 5 + 5. So, 3 x 5 must be the same as -5  5  5. Remember, a double negative is the same as a positive, so 3 x 5 = 15.

Ask: Who can show us a good way to show 15 divided by 3?
Since students have recently worked out the related multiplication, they should realize that they need to start with the quotient and make the proper number of groups. This will also work for 15 divided by 3, for 15 divided by 5, and for 15 divided by 3 or by 5.

Continue with examples that incorporate rational numbers other than integers. help students see that the rules are the same for all rational numbers.

Wrap-Up and Assessment Hints

Rewriting subtraction as addition and thinking about how to show the work on a number line takes lots of practice. When students have spent some time on this, discuss some patterns in the work. If they've had enough practice, and if they're familiar with the concept of absolute value, they should notice:

- When the signs of the addends are the same, the sum has that sign.
- When the signs of the addends are the same, ignore the signs and add to get the sum.
- When the signs of the addends are different, the sum has the sign of the addend whose absolute value is greater.
- When the signs of the addends are different, ignore the signs. Look at the absolute values of both addends. Subtract the smaller absolute value from the larger to get the sum.